


Closure

by eve11



Category: Doctor Who, Doctor Who (2005)
Genre: Gen, Humor, Introspection, Math, Vignette
Language: English
Status: Completed
Published: 2012-09-27
Updated: 2012-09-27
Packaged: 2017-11-15 04:06:03
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 1,018
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/522956
Author URL: https://archiveofourown.org/users/eve11/pseuds/eve11
Summary: <blockquote class="userstuff">
              <p>An argument.  A paperweight.  A bounded infinite space.</p>
            </blockquote>





	Closure

**Author's Note:**

> Written for PhoenixDragon who requested: _The Doctor has a Menger Sponge on the console. River asks about it._ Aw, you know I love a good mathematical metaphor.

**

The TARDIS was still wheezing her way through the last part of materialization when her door was flung open and River sped inside.

"Perfect timing," she said, snapping the door shut and primly smoothing her hair. "I was at the end of my rope."

"Deadly peril, sweetie?" the Doctor looked up from where he'd been studying the console with a frown. "Something I should know about?" 

"Paperwork!" River sashayed up the staircase and put her arms around his waist. "You would not believe the logistics involved in interplanetary expeditions--oh, that's new." Something straight and square on the console had caught her eye. A new addition to the jumble of lights and thingamabobs, that the Doctor had been contemplating when she entered. She peered over his shoulder to get a better look. It was a matte black cube, tessellated with various sized cube-shaped holes. "What is that?"

"Nothing. Just an argument. Never mind." He put his hands over hers and hesitated, only for the span of a second, before turning around in her arms, smiling, and changing the subject. "Perfect timing, now that's new," he said, moving them away. 

River set her feet. "An argument?" she asked, looking back again at the cube. It was, she realized, a perfect cube, about eight inches on each side. The middle of each side had been cut out, as well as the middle of the cube itself. The holes in the surrounding structure of cubes were multi-sized but extremely regular. "How is that--?"

"Oh, you know how she can be when she remembers about concrete dimensions." The Doctor interrupted, waving a hand to encompass the TARDIS central column. "Then before you know it, she starts mixing her metaphors. . . well, it's not important, and anyway, she's wrong."

The TARDIS' background hum deepened for a moment. The Doctor winced and squinted down at the cube. River reached for it, brushing a hand along the top. It was smooth, almost powdery but there was no residue. 

"I think I'd call it a paperweight before calling it an argument. Actually," River paused in thought, and then snapped her fingers. "That's where I've seen it before! It _was_ a paperweight! I had a maths professor had one on her desk. Cubes in cubes in cubes; now I've forgotten the name. I only remember flinging it to the floor before we--" she stopped suddenly, looked in her husband's eyes, and cleared her throat. "Just the once," she said. "Unless you're game to change our plans. This is a Date Night, if I've got our timelines correct."

The Doctor laughed. Whatever mood or argument had taken him, it was dissipating. "It's a three dimensional fractal cube," he said, and raised an eyebrow at her, adding smugly, "and I doubt you've ever seen any one like this before. Take a closer look."

River examined the cube. Her professor's had been tessellated with three different sizes of holes--one set one third the cube's original size, one set one third of that one third--removing the middles of the smaller cubes, making 20 new, even smaller cubes in place of each of the bigger. And a last set one third of the third of the third. Three layers of fractal hole-punching had made a collection of cubes only a centimeter or so in diameter. But this one . . . She squinted. Even at the smallest scale she could see them, tinier and tinier holes, always one-third of the previous size. The image blurred and she blinked.

"How many--?" she said, looking back up at the Doctor.

"Forty-four," he said. "In front of you is a forty-four level fractal lattice. And in the scheme of iterations, forty-four isn't even all that impressive, is it? Tiny number, forty-four."

The TARDIS hummed strangely again. River, being a quick study, did some maths in her head. "That's an octo-decillion . . . that's ten to the fifty-seventh power number of cubes," she said.

The Doctor beamed. "Giving that sponge there a surface area approximately the same size as all of the continents on the Earth… ah, well," he held up a hand, "That's prior to the Great Disasters of year twelve thousand and seventy two. When there were still seven continents."

River stared again. "Those cubes are smaller than the smallest particles in the universe," she said. "That's impossible."

"Maybe for three dimensional beings." The Doctor picked up the cube and tossed it hand to hand. "But she cheats. She can compress the space a bit, letting each step go smaller and smaller--"

"In theory," River said. "But in practice . . . it's a bounded space. If each step is getting smaller and smaller, growing more and more complicated, how much useful surface can you really fit in there?"

"Ad infinitum," the Doctor finished, catching the cube and clunking it solidly back onto the console. "So!" He clapped his hands together and started dancing around the console, pressing buttons and rotating dials. "Date Night! You should get dressed. You've got five minutes."

River set off for the wardrobe. The TARDIS groaned and shuddered and shook, before finally catching her stride into the Vortex.

**

Five minutes and two days later, the TARDIS doors flung inward again. River and the Doctor flew inside, slamming the doors behind them and sagging against them, side by side. His top hat was gone, his suit singed and spotted with green slime. One shoe was missing. Her dress was torn in three places and her hair smelled of energy weapons and ozone. 

"Date night." River stood up, pointing accusingly at the Doctor. "Date night! Again! Oh, my feet are killing me."

"Sorry." The Doctor was still catching his breath at the TARDIS door. "I suppose we shouldn't ever start off an evening in impractical shoes."

"I'm going for a bath." River stormed up the staircase and off toward the bedrooms. "Honestly!" her voice came from the corridors. "How many times can the TARDIS possibly miss the Singing Towers?"

The Doctor sighed, his eyes tracking the space, settling on the dormant Time Rotor.

"That one makes forty-five," he said quietly, and then gathered himself up and followed her.

**


End file.
